In previous chapters we have seen that string theory at the classical level shows promise of describing the Standard Model and can realize at least one scenario for the physics beyond: low-energy supersymmetry. But there are many puzzles, most importantly the existence of moduli and the related question of the cosmological constant. At tree level, in the Calabi– Yau solutions the cosmological constant vanishes. But whether this holds in perturbation theory and beyond requires an understanding of the quantum theory.

In studying string theory, we have certain tools:

1. weak coupling expansions;

2. long-wavelength (low-momentum, α′) expansions.

We have exploited both these techniques already. In analyzing string spectra we worked in a weak coupling limit. There are corrections to the masses and couplings, for example; in string perturbation theory all but a few states that we have studied have finite lifetimes. At weak coupling these effects are small, but at strong coupling the theories will presumably look dramatically different.

In asserting that Calabi–Yau vacua are solutions of the string equations, we used both the above types of expansion. We wrote down the string equations both in lowest order in the string coupling and also with the fewest number of derivatives (two). Even at weak coupling and in the derivative expansion, we can ask whether Calabi–Yau spaces are actually solutions of the string equations, both classically and quantum mechanically. For example, we have seen that, at lowest order in both expansions, there are typically many massless particles. We might expect tadpoles to appear for these fields, both in the α′ expansion and in loops. There is in general no guarantee that we can find a sensible solution by simply perturbing the original one.

Yet there are many cases where we can make exact statements. In both Type II and heterotic string theories, we can often show that Calabi–Yau vacua correspond to exact solutions of the classical string equations. We can also show that they are good vacua – there are no tadpoles for massless fields – to all orders of the string perturbation expansion. More dramatically, we can sometimes show that these vacua are good, nonperturbative, states of the theory. This is perhaps surprising since we lack a suitable nonperturbative formulation in which to address this question directly. The key to this magic is supersymmetry.

In the previous chapter we were forced to face the fact that on the one hand string theory, if it describes nature, is not weakly coupled. On the other hand, the very formulation of the theory that we have put forward is perturbative. We have described the quantum mechanics of single strings and given a prescription for calculating their interactions order by order in perturbation theory in a parameter gs. There is a parallel here to Feynman's early work on relativistic quantum theory: Feynman guessed a set of rules for computing the perturbative amplitudes of electrons. In that case, however, one already had a candidate for an underlying description: quantum electrodynamics. It was Dyson who clarified the connection. For Abelian theories a non-perturbative approach probably does not exist, but in the case of non-Abelian gauge theories it does. The field-theoretic formulation provides an understanding of the underlying symmetry principles and access to a treasure trove of theoretical information.

A string field theory would be a complicated object. The string fields themselves would be functionals of the classical two-dimensional fields which describe the string. The quantization of such fields is sometimes called the “third quantization.” Much effort has been devoted to writing down such a field theory. For open strings one can obtain relatively manageable expressions which reproduce string perturbation theory. For closed strings, infinite sets of contact interactions are required. But, quite apart from their cumbersome structure, there are reasons to suspect that this is not a useful formulation. There would seem to be, for example, vastly too many degrees of freedom. At one loop we have seen that the string amplitudes are to be integrated only over the fundamental region the moduli space. Naively, a field theory which simply describes all of the states of the string would have amplitudes integrated over the whole region, and the cosmological constant would be extremely divergent. The contact interaction terms mentioned above solve this problem but not in a very satisfying way.

Despite this, there has been great progress in understanding the non-perturbative aspects of the known string theories. Most strikingly, it is now known that all theories with 16 or more supersymmetries are the same. Many tools have been developed to study phenomena beyond string perturbation theory, especially D-branes and supersymmetry.

The LHC, in its first years of running, has been a remarkable success. The discovery of the Higgs boson in an extremely complex environment is an extraordinary achievement, both experimentally and also in the application of our understanding of many facets of the Standard Model. This particle appears, at the 10%−20% level, in several channels, to be the Higgs field of the simplest version of the Standard Model. Over the next few years these measurements will improve and additional channels will be studied. In Chapter 4 of this text we studied the Standard Model as an effective-field theory. In that discussion our treatment of the Higgs sector was somewhat tentative; we entertained the possibility that the Standard Model might fail at scales of order 1 TeV. It is quite possible, however, that over the next few years we will establish that the Standard Model, including only a single Higgs doublet, provides a complete description of nature up to a scale of a few TeV. This would represent an extraordinary achievement.

Yet we have many unanswered questions. As this book goes to press the LHC is beginning to run at close to its design energy of 14 TeV. It is quite possible that, as we explore this new energy frontier, we will see one or more major discoveries – a candidate for dark matter, evidence for supersymmetry, additional Higgs fields, a new U(1) gauge boson Z′ or something totally unanticipated. Experiments at the cosmic frontier searching for dark matter, CMB polarization or non-Gaussianity and other phenomena are coming on line and/or improving their reach, and major discoveries might be made over the next few years. Alternatively we have seen that the LHC has already excluded many possibilities for new physics. It is conceivable that the answers to many questions do not lie at energies which will be accessible in the next few years.

To conclude this book, an assessment of some of the ideas for Beyond the Standard Model physics, and their prospects, is appropriate.

The hierarchy or naturalness problem

The hierarchy problem is strongly suggestive of new physics at TeV energy scales. Supersymmetry, broken at around one TeV, is a possible solution which we have explored extensively in this book.

In a standard advanced field theory course, one learns about a number of symmetries: Poincaré invariance, global continuous symmetries, discrete symmetries, gauge symmetries, approximate and exact symmetries. These latter symmetries all have the property that they commute with Lorentz transformations and in particular with rotations. So, the multiplets of the symmetries always contain particles of the same spin; in particular, they always consist of either bosons or fermions.

For a long time, it was believed that these were the only allowed types of symmetry; this statement was even embodied in a theorem, known as the Coleman–Mandula theorem. However, physicists studying theories based on strings stumbled on a symmetry which related fields of different spin. Others quickly worked out simple field theories with this new symmetry, called supersymmetry.

Supersymmetric field theories can be formulated in dimensions up to eleven. These higher-dimensional theories will be important when we consider string theory. In this chapter we consider theories in four dimensions. The supersymmetry charges, because they change spin, must themselves carry spin – they are spin-1/2 operators. They transform as doublets under the Lorentz group, just like the two-component spinors χ and χ*. (The theory of two-component spinors is reviewed in Appendix A, where our notation, which is essentially that of the text byWess and Bagger (1992), is explained.) There can be 1, 2, 4 or 8 such spinors; correspondingly, the symmetry is said to be N = 1, 2, 4 or 8 supersymmetry. Like the generators of an ordinary group, the supersymmetry generators obey an algebra; unlike an ordinary bosonic group, however, the algebra involves anticommutators as well as commutators (it is said to be “graded”).

There are at least four reasons to think that supersymmetry might have something to do with TeV-scale physics. The first is the hierarchy problem: as we will see, supersymmetry can both explain how hierarchies arise, and why there are no large radiative corrections. The second is the unification of couplings. We have seen that while the gauge group of the Standard Model can in a rather natural way be unified in a larger group, the couplings do not unify properly. In the minimal supersymmetric extension of the Standard Model (the minimal supersymmetric Standard Model, or MSSM) the couplings unify nicely if the scale of supersymmetry breaking is about 1 TeV.

We have focused in this text on several questions of naturalness, and have used them to motivate searches for possible new physics. It is fair to say that most physicists find this principle compelling and are reluctant to accept extreme (or even modest!) fine tunings in theories of natural phenomena. But, during the past decade, a plausible, if highly speculative, alternative picture has gained currency, known as the landscape. If correct it provides a picture for the emergence of the laws of nature in which fine tunings are not surprising and provide few or no clues as to new degrees of freedom that might lie at higher energy scales.

We will divide our discussion into two parts. First we will explain, in very general terms, what is meant by a landscape and how it might address some naturalness problems in our current understanding of particle physics. Then we consider models for how a landscape might arise in string theory. These models are at best plausible; the existence of any nonsupersymmetric states in string theory (apart, possibly, from certain special AdS vacua), much less vast numbers of them, is hardly established.

The cosmological constant revisited

We have stressed that the cosmological constant (i.e. the dark energy) presents potentially the most striking failure of naturalness. One might hope to solve this problem by introducing new degrees of freedom. Supersymmetry helps to some extent. In global supersymmetry the ground state energy is well defined and of order the scale of supersymmetry breaking raised to the fourth power. In local supersymmetry there is also the term -3|W|2 in the potential. The problem is that this last term must very nearly cancel the positive contributions from supersymmetry breaking. The superpotential W can naturally be small as a result of R symmetries, but no one has proposed a mechanism, based on either dynamics or symmetries, which would lock W onto its required value. Many physicists have searched for an analog of the axion solution of the strong CP problem, in which some light field would adjust in such a way as to cancel the c.c. Without reviewing the various proposals, one might expect that the basic obstacle is in fact illustrated by the Peccei– Quinn mechanism.

As this is being written, particle physics stands on the threshold of a new era, with the commissioning of the Large Hadron Collider (LHC) not even two years away. In writing this book, I hope to help prepare graduate students and postdoctoral researchers for what will hopefully be a period rich in new data and surprising phenomena.

The Standard Model has reigned triumphant for three decades. For just as long, theorists and experimentalists have speculated about what might lie beyond. Many of these speculations point to a particular energy scale, the teraelectronvolt (TeV) scale, which will be probed for the first time at the LHC. The stimulus for these studies arises from the most mysterious – and still missing – piece of the Standard Model: the Higgs boson. Precision electroweak measurements strongly suggest that this particle is elementary (in that any structure is likely to be far smaller than its Compton wavelength), and that it should be in a mass range where it will be discovered at the LHC. But the existence of fundamental scalars is puzzling in quantum field theory, and strongly suggests new physics at the TeV scale. Among the most prominent proposals for this physics is a hypothetical new symmetry of nature, supersymmetry, which is the focus of much of this text. Others, such as technicolor, and large or warped extra dimensions, are also treated here.

Even as they await evidence for such new phenomena, physicists have become more ambitious, attacking fundamental problems of quantum gravity and speculating on possible final formulations of the laws of nature. This ambition has been fueled by string theory, which seems to provide a complete framework for the quantum mechanics of gauge theory and gravity. Such a structure is necessary to give a framework to many speculations about Beyond the Standard Model physics. Most models of supersymmetry breaking and theories of large extra dimensions or warped spaces cannot be discussed in a consistent way otherwise.

It seems, then, quite likely that a twenty-first-century particle physicist will require a working knowledge of supersymmetry and string theory, and in writing this text I hope to provide this. The first part of the text is a review of the Standard Model.

While perturbation theory is a powerful and useful tool in understanding field theories, for our exploration of physics beyond the Standard Model an understanding of nonperturbative physics will be crucial. There are many reasons for this.

1. One of the great mysteries of the Standard Model is non-perturbative in nature: the smallness of the θ parameter.

2. Strongly interacting field theories will figure in many proposals to understand other mysteries of the Standard Model.

3. The interesting dynamical properties of supersymmetric theories, both those directly related to possible models of nature and those which provide insights into broad physics issues, are non-perturbative in nature.

4. If string theory describes nature, non-perturbative effects are necessarily of critical importance.

We have introduced lattice gauge theory, which is perhaps our only tool for doing systematic calculations in strongly coupled theories. But, as a tool, its value is quite limited. Only a small number of calculations are tractable in practice, and the difficult numerical challenges sometimes obscure the underlying physics. Fortunately, there is a surprising amount that one can learn from symmetry considerations, from semiclassical arguments and from our experimental knowledge of one strongly coupled theory, QCD. In each of these, an important role is played by the phenomena known as anomalies and, related to these, a set of semiclassical field configurations known as instantons.

Usually, the term “anomaly” is used to refer to the quantum mechanical violation of a symmetry which is valid classically. Instantons are finite-action solutions of the Euclidean equations of motion, typically associated with tunneling phenomena. Anomalies are crucial to understanding the decay of the π0 in QCD. Anomalies and instantons account for the absence of a ninth light pseudoscalar meson in the hadron spectrum. Within the weak-interaction theory, anomalies and instantons lead to violations of baryon and lepton number; these effects are unimaginably tiny at the current time but were important in the early universe. The absence of anomalies in gauge currents is important to the consistency of theoretical structures, including both field theories and string theories. The cancelation of anomalies within the Standard Model itself is quite non-trivial. Similar constraints on possible extensions of the Standard Model will be very important.

While motivated in part by the hopes of building phenomenologically successful models of particle physics, we have uncovered in our study of supersymmetric theories a rich treasure trove of field theory phenomena. Supersymmetry provides powerful constraints on the dynamics. In this chapter we will discover more remarkable features of supersymmetric field theories. We will first study classes of (super)conformally invariant field theories. Then we will turn to the dynamics of supersymmetric QCD with Nf ≥ Nc, where we will encounter new, and rather unfamiliar, types of behavior.

Conformally invariant field theories

In quantum field theory, theories which are classically scale invariant are typically not scale invariant at the quantum level. Quantum chromodynamics is a familiar example. In the absence of quark masses we believe that the theory predicts confinement and has a mass gap. The CPN models are an example where we were able to show systematically how a mass gap can arise in a scale-invariant theory. In all these cases the breaking of scale invariance is associated with the need to impose a cutoff on the high-energy behavior of the theory. In a more Wilsonian language one needs to specify a scale where the theory is defined, and this requirement breaks the scale invariance.

There is, however, a subset of field theories which are indeed scale invariant. We have seen this in the case of N = 4 supersymmetric field theories in four dimensions. In this section we will see that this phenomenon can occur in N = 1 theories and will explore some of its consequences. In the next section we will discuss a set of dualities among N = 1 supersymmetric field theories, in which conformal invariance plays a crucial role.

In order that a theory exhibit conformal invariance it is necessary that its beta function vanish. At first sight it would seem difficult to use perturbation theory to find such theories. For example, one might try to choose the number of flavors and colors in such a that the one-loop beta function vanishes. But then the two-loop beta function will generally not vanish.